The spectral index \(n_s\)  and amplitude \(A\)  at CMB scales should be consistent with the CMB observations [1]}, but we have more freedom to change the perturbations at small (PBH) scales.
As can be read from this equation, we allow variation of both the tilt \(\tilde{n}_s\)  of the power spectrum at PBH scales and its running \(\tilde{\alpha }_s\) . Models such as critical Higgs inflation can indeed be better represented by a non-vanishing running of the spectral index [2]}, [3]}. In the remaining part of this Section, we will take the fiducial values to be \(\tilde{n}_s=0.97\)  and \(\tilde{\alpha }_s=0\) , and analyze their variation in the following Section. The amplitude \(\tilde{A}\)  is determined by fixing \(f_{\rm PBH}^{\rm tot}=1\)  using Eq. (REF ).
